If there is a Floppy based version of the BIOS update, use that (if you have to, you may need to buy a USB based floppy drive or barrow a friend's).
Other ideas include creating a BartPE/UltimateBoot CD for Windows (UBCD4WIN)/Windows PE CD and trying to install from that.
Is there a specific reason you are updating the BIOS? When it comes to BIOS, I GENERALLY follow the thinking that "if it ain't broke, don't fix it" - if you need some new functionality of the new BIOS, that's one thing, but if you're upgrading just because there's an upgrade, at least with BIOS, I don't consider that wise.
Reference Links:
EDIT: Also remembered - there are boot CDs you can make - DOS boot CDs that can skip the floppy - I've only done it once and it worked ok. Check out www.bootdisk.com (but just remember, you need a DOS version of the BIOS updater for that to work)
And just to clarify - 64 bit versions of Windows have ZERO support for 16 bit applications, so that 16 bit app will never run. If there's a 32 bit version, that MIGHT work, but apps that interact directly with hardware like that often NEED to be 64 bit on a 64 bit system. (Although I suppose if a 16 bit BIOS update app works on a 32 bit system, then a 32 bit BIOS update app might just work fine on a 64 bit system)
Warning: Because BIOS flashing is potentially risky, if you do not encounter problems using the current version of BIOS, it is recommended that you not flash the BIOS. If you want to flash the BIOS, do it with caution. Inadequate BIOS flashing may result in system malfunction. – None – 2011-02-18T12:00:27.920
Have you tried a DOS boot disk? I think FLASHSPI can run from DOS. – hyperslug – 2009-09-21T19:38:58.007